So many great choices, how does one decide? I would think a lot about food. All the European regions discussed have distinctive wonderful local cuisines. If you’ve never been to Europe you’ll be shocked how good the meals are, and how reasonably priced. Pick your favorite cuisine, then go see how perfectly the local wines match it 
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Forget the prices of the wines. You can get any wine you want in NYC. For your first trip to a wine region, go to one of the classics - Bordeaux, Burgundy, Piedmont, Tuscany, or something like that. You say you like to travel and that you are new to wine. The trip is about the experience. It isn’t a wine purchasing trip. How much are you going to lug back with you anyway.
The classics are classics for a reason. Think of this as a travel trip. Would it be exciting to even drive by Chateau Margaux, Chateau Latour, etc. Or to park by the vineyard of Romanee Conti or Chambertin or something. I still need to visit the Italian wine regions, but people tell me Barolo is breathtakingly beautiful. And, Tuscany is Tuscany. From NY, flying to France or Italy is just as easy as flying to SF or Oregon. From Paris, it is 1 1/2 hours by TGV to Dijon.
You may get a $30 wine for $15 in some obscure wine region, but wouldn’t you rather have meals in one of the classic regions. I first went to a wine region in 1984. My wife and I (pre-kids) were in Paris and I got a couple of days in Burgundy. I could not believe I was there. Driving through the vineyards, walking through the towns, eating the food, one great experience after another. Go to a region that you will remember the rest of your life.
If you are really looking for quick and easy from New York, skip the Finger Lakes and instead go to Niagara-on-the-Lake, Ontario; it fits the bill. I can’t wait to go back to Pearl Morissette.
Wine country visits can be loads of fun, especially when just starting out.
I can not tell what your objective function is here when you say on a budget. The cost to travel to and stay at a distant location will be by far the largest part of the expense. Any wine you may buy will be a tiny portion and maybe zero. Most wine regions can be fairly well priced to stay in and visit with the exception of Napa. Maybe Bordeaux but I have not been there. So really if your objective is to visit a great wine area on a budget that leaves most everything open. If your objective it is to go someplace to buy wine (not ideal IMHO) that is usually an unneeded hassle when traveling. In the US it is easy to have a winery ship back to you. Overseas it is really simplest to buy the wines you like stateside but you can check a wine shipper for special bottles that are unlikely to be easily found here and check it with luggage.
And unbeknownst to most Americans, Canada is actually a foreign country as well. Clearly. if you are looking for inexpensive foreign travel, This should be the number one spot on your list.
Echoing some of the above, I would say there are two broad choices here – visit a classic wine region. You will then get a feel for the geography, the culture, the cuisine. It will help inform how you think about the wines. So Howard’s comments above about Bordeaux, Burgundy, Piedmont, Tuscany, etc. Second is to go some place to be exposed to a completely different set of wines. Croatia, Slovenia, etc. In this category I know South Africa well. Cape Town is breathtakingly beautiful (though generally bad weather their winter which is our summer – though that makes things cheaper). Yes, wines will be cheaper there than here – but as many note that is not a good reason to go. Instead there are wines available in South Africa (and thus I suspect in these other places) that are not available in the US. That is of course also true for the classic regions, but for this second group I would emphasize expanding your knowledge of a new set of wines, rather than deepening your knowledge of a classic area.
Just my two cents.
Sorry, Canada is the 51st state of the U.S. Condolences if you thought you were a sovereign nation.
Trite and ignorant Marcus.
I guess the real question is how much of a budget are we talking?? From NYC one could drive to Alba winery in New Jersey or to the Finger Lakes.
Airfare from NYC can be a bargain so trips to wine country in Germany, Italy, Spain, southern France, and Portugal can all be relative bargains.It’s odd how inexpensive Germany can be.
The Piedmont is a bargain in some ways. We stayed at a very nice hotel with one of the best hotel breakfasts (free) I’ve ever had. 110 Euros a night. A truffle feast with a bottle of Arneis and two barolos came to 600 euros for four. When you consider what that meal would have cost elsewhere…
Friends just came back from Argentina…they thought things cost 25% of what they cost here.
Any thoughts, OP?
Niagara on the Lake is beautiful with very good Pinot noir and excellent ice wine. Most beautiful wine region in all of North America is the Okanagan in British Columbia which gone through explosive growth with perhaps 250+ wineries now. Base yourself in Kelowna. Current exchange rate is US 1.35 to CD 1.00. I’d also make a serious argument for visiting Vancouver which is about four hours away.
If you make it to California, there are a few regions that are budget friendly (compared to Napa) with great scenery and side trips. All of these would require some driving from SFO.
- Amador
- Mendocino
- Lodi
- Livermore
- Paso Robles
- San Luis Obispo (Edna Valley)
- Santa Cruz Mountains
As destinations yes, getting there not so much. Have not seen good plane fares to either place lately.
Another question to think about is how open are wineries for visitors??
Cognac and Champagne are set up for visits because they know there is a good chance you can buy their products back home. Visiting wineries in Burgundy is not easy because wineries don’t make much wine.
I have no idea about visiting wine regions as a tourist.